Gorenstein Projective Dimensions of Modules over Minimal Auslander–Gorenstein Algebras
Gorenstein Projective Dimensions of Modules over Minimal Auslander–Gorenstein Algebras
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DOI：
10.1142/s1005386721000262
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发表时间：
2018-02
影响因子：
0.3
通讯作者：
Shen Li;René Marczinzik;Shunhua Zhang
中科院分区：
文献类型：
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作者：
Shen Li;René Marczinzik;Shunhua Zhang
本文研究了极小Auslander-Gorenstein代数的[公式：见文本]-模的Gorenstein射影维度与它们的基座之间的关系。首先，我们给出了投射-内射[公式：见正文]-模的基座的描述。然后证明了[公式：见文本]-模[公式：见文本]至多有[公式：见文本]-模的Gorenstein投射维度当且仅当它的基座至多有[公式：见文本]-模的Gorenstein射影维度当且仅当[公式：见文本]-模是由投射的[公式：见文本]-模上生成的。此外，我们还证明了[公式：见正文]-极小Auslander-Gorenstein代数可以由模的Gorenstein射影维度与其基座之间的关系来刻画。
In this article we investigate the relations between the Gorenstein projective dimensions of [Formula: see text]-modules and their socles for [Formula: see text]-minimal Auslander–Gorenstein algebras [Formula: see text]. First we give a description of projective-injective [Formula: see text]-modules in terms of their socles. Then we prove that a [Formula: see text]-module [Formula: see text] has Gorenstein projective dimension at most [Formula: see text] if and only if its socle has Gorenstein projective dimension at most [Formula: see text] if and only if [Formula: see text] is cogenerated by a projective [Formula: see text]-module. Furthermore, we show that [Formula: see text]-minimal Auslander–Gorenstein algebras can be characterised by the relations between the Gorenstein projective dimensions of modules and their socles.